GATE CSE 2007 | Question: 21

Question

How many different non-isomorphic Abelian groups of order $4$ are there?

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Comments

Detailed Video Explanation: https://www.youtube.com/live/Zb4OWjHL3yU?t=2048&feature=shared

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Groups of Small Orders: https://www.youtube.com/live/Zb4OWjHL3yU?t=231&feature=shared

Some Important Points about Abelian Groups: https://www.youtube.com/live/Zb4OWjHL3yU?t=2180&feature=shared

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When you are makink caylay table, then one structure where two elements have different inveres, and other tructire with same inverse

 

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Note : Important result 

Every group of order ≤ 4 is abelian.

Every group of prime order is abelian.

Therefore, every group of order ≤ 5 is abelian.

The smallest non-abelian group has order 6 (specifically, S3​).

Every group of prime order is Abelian

Proof: Let G be a group of prime order p. Pick any non-identity element a∈G (such an element exists because p ≥ 2). By Lagrange's theorem, the order of the cyclic subgroup ⟨a⟩ must divide |G| = p. Since a ≠ e , the order of a is not 1, so it must be exactly p. Therefore, ⟨a⟩ has p elements, which means it is the entire group G. Hence, G is cyclic.

 

Answer 1

The number of Abelian groups of order  $P^{k}$ ($P$ is prime) is the number of partitions of  $k.$
Here, order is $4$ i.e. $2^{2}$.
Partition of $2$ are $\{1,1\}, \{2,0\}.$
Total $2$ partition so no. of different abelian groups are $2.$

http://oeis.org/wiki/Number_of_groups_of_order_n

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1. First, find the prime factorization of $n.$ For example, $4$ has prime factorization as $2*2.$ Also, $600$ can be factorized as $2^3∗3^1∗5^2$
2. Now, find the number of partitions of all powers, and then multiply them. Number of Partitions of a number $k$ is the number of ways $k$ can be partitioned. For example, number of partitions of $3$ is $3,$ because $3$ can can be partitioned in $3$ different ways: $\{1+1+1\}, \{1+2\}, \{3\}.$ Similarly, $4$ can be partitioned in $5$ different ways: $\{1+1+1+1\},\{2+1+1\},\{2+2\},\{3+1\},\{4\}.$ Note that order of elements in a partition does not matter, for example, partitions $\{2+1+1\}$ and $\{1+1+2\}$ are the same.  
   So for this question, we will find number of partitions of $2,$ which is $2 : \{1+1\},\{2\}.$ There is no other power, so answer is $2$ only.
3. Suppose, in question, order given is $600.$. Then, different powers are $3,1,2$. Number of partitions for $3,1,2$  are $3,1,2$  respectively and result would have been $3∗1∗2=6.$
4. Group theorists view two isomorphic groups as follows: For every element g of a group G, there exists an element h of H  group such that h 'behaves in the same way' as g(operates with other elements of the group in the same way as g). For instance, if g generates G, then so does h. This implies in particular that G and H are in bijective correspondence. Thus, the definition of an isomorphism is quite natural.

Correct Answer: $A$

Comments

very nice explaination sir

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Must Read this from Joseph A. Gallian book 

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The number of non-isomorphic abelian groups of order p^n (p = prime no)  is exactly equal to the number of partitions of n 

Order = 4 ⇒ Prime factorization = 2^2

Let n=2. We need to find all integer partitions of the number 2 :

Partition 1: 2 (just itself)

Partition 2: 1+1

Since there are 2 partitions, there are exactly 2 non-isomorphic abelian groups of order 4.

Answer 2

Answer: A

Number of different non-isomorphic abelian groups of order n are $\prod_{i=1}^{w(n)} p(a_i)$, where $p(a_i)$ is the number of partitions of the i^th prime number. Here the order is only divisible by 2 and number of partitions of 2 are 2: {0,2} and {1,1}.

Hence, the answer is 2.

Answer 3

Watch this

Start from 6:34

 

Answer 4

The answer is Option A (only two partition )

Answer 5

Option b .

Comments

1. First, find the prime factorization of n. For example, 4 has prime factorization as 2*2. Also, 600 can be factorized as 2^(3)∗3^(1)∗5^(2)
2. Now, find the number of partitions of all powers, and then multiply them. Number of Partitions of a number k is the number of ways k can be partitioned. For example, number of partitions of 3 is 3, because 3 can can be partitioned in 3 different ways : {1+1+1}, {1+2}, {3}. Similarly, 4 can be partitioned in 5 different ways : {1+1+1+1},{2+1+1},{2+2},{3+1},{4}. Note that order of elements in a partition doesn't matter, so for example, partitions {2+1+1} {1+1+2} are same. 
   So for this question, we will find number of partitions of 2, which is 2 : {1+1},{2}. There is no other power, so answer is 2 only.
3. Suppose, in question, order given is 600, , so different powers are 3,1,2. Number of partitions for 3,1,2  are 3,1,2  respectively, so result would have been 3∗1∗2=6.

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what is meant by isomorphic and non-isomorphic abelian group,did't get any good reference.

@Kaluti any help?

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group theorists view two isomorphic groups as follows: For every element g of a group G, there exists an element h of H  group such that h 'behaves in the same way' as g(operates with other elements of the group in the same way as g). For instance, if g generates G, then so does h. This implies in particular that G and H are in bijective correspondence. Thus, the definition of an isomorphism is quite natural. 

  you can refer this